Question

Three waves of amplitude $10\mu m,4\mu m$ and $7\mu m$ arrive at a point with successive phase difference of $\pi /2$. The amplitude of the resultant wave is

• A. $2\mu m$
• B. $7\mu m$
• C. $5\mu m$
• D. $1$

Answer 1

Answer: (C)

$5\mu m$

Given,

Amplitude of the waves are,

$a_1=10\mu m,a_2=4\mu m,a_3=7\mu m$

and the phase difference between 1st and 2nd wave is $\frac{\pi }{2}$ and that between 2nd and 3rd wave is $\frac{\pi }{2}$. Therefore, phase difference between 1st and 3rd is $\pi$.

Combining 1st with 3rd, their resultant amplitude will be

$A_1^2=a_1^2+a_3^2+2a_1{a}_{3}cos\varphi$

or

$A_1=\sqrt{{10}^2+7^2+2\times 10\times 7cos\pi }$

$=\sqrt{100+49-140}=\sqrt{9}=3\mu m$ in the direction of first.

Now combining this with 2nd wave we get the resultant wave,

$A^2=A_1^2+a_2^2+2A_1{a}_2\frac{cos\pi }{2}$

or

$A=\sqrt{3^2+4^2+2\times 3\times 4cos{90}^2}=\sqrt{9+16}=5\mu m$